What is it, pray tell, that Omega cannot do?
Can he not scan your brain and determine what strategy you are following? That would be odd, because this is no stronger than the original Newcomb problem and does not seem to contain any logical impossibilities.
Can he not compute the strategy, S, with the property "that at each moment, acting as S tells you to act -- given (1) your beliefs about the universe at that point and (2) your intention of following S at all times -- maximizes your net utility [over all time]?" That would be very odd, since you seem to believe a regular person can compute S. If you can do it, why not Omega? (NB, no, it doesn't help to define an approximation of S and use that. If it's rational, Omega will punish you for it. If it's not, why are you doing it?)
Can he not compare your strategy to S, given that he knows the value of each? That seems odd, because a pushdown automaton could make the comparison. Do you require Omega to be weaker than a pushdown automaton?
No?
Then is it possible, maybe, that the problem is in the definition of S?
What is it, pray tell, that Omega cannot do?
Well, for instance, he cannot make 1+1=3. And, if one defines rationality as actually winning then he cannot act in such a way that rational people lose. This is perfectly obvious; and, in case you have misunderstood what I wrote (as it looks like you have), that is the only thing I said that Omega cannot do.
In the discussion of strategy S, my claim was not about what Omega can do but about what you (a person attempting to implement such a strategy) can consistently include in your model of the universe. If yo...
Followup to: Newcomb's Problem and Regret of Rationality
"Rationalists should win," I said, and I may have to stop saying it, for it seems to convey something other than what I meant by it.
Where did the phrase come from originally? From considering such cases as Newcomb's Problem: The superbeing Omega sets forth before you two boxes, a transparent box A containing $1000 (or the equivalent in material wealth), and an opaque box B that contains either $1,000,000 or nothing. Omega tells you that It has already put $1M in box B if and only if It predicts that you will take only box B, leaving box A behind. Omega has played this game many times before, and has been right 99 times out of 100. Do you take both boxes, or only box B?
A common position - in fact, the mainstream/dominant position in modern philosophy and decision theory - is that the only reasonable course is to take both boxes; Omega has already made Its decision and gone, and so your action cannot affect the contents of the box in any way (they argue). Now, it so happens that certain types of unreasonable individuals are rewarded by Omega - who moves even before they make their decisions - but this in no way changes the conclusion that the only reasonable course is to take both boxes, since taking both boxes makes you $1000 richer regardless of the unchanging and unchangeable contents of box B.
And this is the sort of thinking that I intended to reject by saying, "Rationalists should win!"
Said Miyamoto Musashi: "The primary thing when you take a sword in your hands is your intention to cut the enemy, whatever the means. Whenever you parry, hit, spring, strike or touch the enemy's cutting sword, you must cut the enemy in the same movement. It is essential to attain this. If you think only of hitting, springing, striking or touching the enemy, you will not be able actually to cut him."
Said I: "If you fail to achieve a correct answer, it is futile to protest that you acted with propriety."
This is the distinction I had hoped to convey by saying, "Rationalists should win!"
There is a meme which says that a certain ritual of cognition is the paragon of reasonableness and so defines what the reasonable people do. But alas, the reasonable people often get their butts handed to them by the unreasonable ones, because the universe isn't always reasonable. Reason is just a way of doing things, not necessarily the most formidable; it is how professors talk to each other in debate halls, which sometimes works, and sometimes doesn't. If a hoard of barbarians attacks the debate hall, the truly prudent and flexible agent will abandon reasonableness.
No. If the "irrational" agent is outcompeting you on a systematic and predictable basis, then it is time to reconsider what you think is "rational".
For I do fear that a "rationalist" will clutch to themselves the ritual of cognition they have been taught, as loss after loss piles up, consoling themselves: "I have behaved virtuously, I have been so reasonable, it's just this awful unfair universe that doesn't give me what I deserve. The others are cheating by not doing it the rational way, that's how they got ahead of me."
It is this that I intended to guard against by saying: "Rationalists should win!" Not whine, win. If you keep on losing, perhaps you are doing something wrong. Do not console yourself about how you were so wonderfully rational in the course of losing. That is not how things are supposed to go. It is not the Art that fails, but you who fails to grasp the Art.
Likewise in the realm of epistemic rationality, if you find yourself thinking that the reasonable belief is X (because a majority of modern humans seem to believe X, or something that sounds similarly appealing) and yet the world itself is obviously Y.
But people do seem to be taking this in some other sense than I meant it - as though any person who declared themselves a rationalist would in that moment be invested with an invincible spirit that enabled them to obtain all things without effort and without overcoming disadvantages, or something, I don't know.
Maybe there is an alternative phrase to be found again in Musashi, who said: "The Way of the Ichi school is the spirit of winning, whatever the weapon and whatever its size."
"Rationality is the spirit of winning"? "Rationality is the Way of winning"? "Rationality is systematized winning"? If you have a better suggestion, post it in the comments.